Annual SPY returns

SPY = S&P 500 ETF

Compounded SPY returns

value of $1 investment with dividends reinvested

Compounded returns on log scale: motivation

• Let’s look at accumulations from two hypothetical stocks.
  • stock 1: 10% per year
  • stock 1: 2% per year until 2000 and 10% afterwards
• It will appear that stock 2 did nothing before 2000 and earned a lot less than stock 1 even after 2000.

Plot of the Example

Log (base 10) of accumulation

Map \(y\) tick labels to dollars

Compounded SPY returns on log scale

value of $1 investment with dividends reinvested

Box Plot

• Box contains 25th percentile through 75th percentile

• Median is indicated as a line in the box

• Fences extend 1.5 times inter-quartile range from 25th and 75th percentiles or to the most extreme observation if that is closer to the box

  • inter-quartile range = 75th minus 25th percentile)

• Points outside the fences are outliers

  • if you simulate data from a normal distribution, there will typically be very few points outside the fences

Box and density plots of annual SPY returns

Autocorrelations

• Autocorrelation is the correlation of a time series with its own lagged values.
• Autocorrelation at lag 1 tells us whether the current value predicts the next one.
• For monthly data, autocorrelation might be high at lag 12 (seasonality).

Autocorrelations of annual SPY returns

Does this year’s return predict next year’s?

No, the autocorrelation is almost zero.